Moskva, Moscow, Russian Federation
Russian Federation
Russian Federation
The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.
geometry, descriptive geometry, loci, L, analytical geometry.
Введение
Изучением геометрических мест точек первым занимался в 1941 г. Дмитрий Иванович Каргин (1880–1949) [13; 18; 23]. Изучали геометрические места точек Александр Давидович Посвянский (1909–…) с коллегами [24], Владимир Яковлевич Волков (1946–2017) с коллегами [2; 3], Геннадий Сергеевич Иванов [14–16], он же с коллегами [28; 29], Антон Георгиевич Гирш [9], Н.В. Наумович [21], И.И. Александров [1]. Совсем недавно нам удалось установить — изучал равноудаленные геометрические места в конце 50-х — начале 60-х гг. прошлого века В.В. Глоговский [10–12], особенно отметим его статью «Эквидистанты» [10]. В это же время вышла книга Н.В. Наумович «Геометрические места в пространстве» [21]. Затронули тему геометрических мест точек Марк Яковлевич Выгодский (1898–1965) в своих ставших классическими справочниках по элементарной и высшей математике и в работе [4], а также один из авторов этой публикации [6–8]. На Всероссийском студенческом конкурсе «Инновационные разработки» за одиннадцать лет его существования было заслушано 59 проектов самой разной тематики: 3D-моделирование, элементы САПР, подвижной состав железных дорог, двигатели, турбины, компрессоры и их части, геометрия, энергосберегающие установки, автомобили и другие передвижные средства, сигнализация, солнечные часы, строительство, история науки и техники, методические вопросы преподавания и пр. [5]. Пять из них (8%) — работы по равноудаленным геометрическим местам. Почти все проекты завоевали призовые места. В 2013–2014 гг. — третьи места, в 2017-м — второе и в 2018-м один из авторов этой работы завоевал третье место. Данная статья является продолжением работ «Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 1» [6] и «Геометрические места точек, равноотстоящих от двух заданных геометрических фигур. Часть 2: геометрические места точек, равноудаленных от точки и конической поверхности» [7]. В предлагаемой вашему вниманию работе рассматриваются ГМТ, равноудаленных от: 1) сферы и прямой; 2) конической поверхности и плоскости. Основой для систематизации ГМТ является табл. 1, приведенная в работе [6].
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